168 VIII DIFFERENTIAL CALCULUS

Here again, to apply (8.5.1) we have only to verify that £ -* ||/(£)ll is regulated.
Finally, we express for integrals results corresponding to (8.6.4) and
(8.6.5):

(8.7,8) If a sequence (gn) of regulated functions, defined in a compact interval
I = [a, /?], converges uniformly in I to g, then the sequence y a gn(£) d£) con-
verges to (^#(0 d%. (Remember g is regulated by (7.6.1).)

(8.7.9) If a series («„) of regulated functions, defined in a compact interval

00

I = [a, /?]» & normally convergent (Section 7.1) in I, then, if u = ]T w«>

n = Q

I*B

the series of general term un(Qd£ is absolutely convergent, and

Jo.

The absolute convergence follows at once from the assumption and the
mean value theorem (8.7.7).

(87.10) Remark. Due to (8.6.4) and the proof of (7.6.1), for any regulated
function /defined in [a, /?], and for any e > 0, there is an increasing sequence

such that

n-l

Ja

If /is continuous, one may (due to (3.16.5)) take all numbers xfc+1 — xk
equal to (j3 - a)//i, and rfc = xk (see Problem 1).

PROBLEMS

1. Let / be a regulated function defined in a compact interval I ^ R. Show that
for any e>0, there is a number S>0 such that for any increasing sequence
*o =s£ t0 ^ xi ^ - • - ^ xk < 4 ^ xk+1 < • - • < xn of points of I for which xk+1 - xk ** 8,
we have

("Riemann sums"; consider first the case in which/is a step-function). G. Then